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MEASURING BUSINESS CYCLES INTRA-SYNCHRONIZATION IN US: A REGIME-SWITCHING INTERDEPENDENCE FRAMEWORK

Danilo Leiva-Leon

Documentos de Trabajo N.º 1726

2017

MEASURING BUSINESS CYCLES INTRA-SYNCHRONIZATION IN US:

A REGIME-SWITCHING INTERDEPENDENCE FRAMEWORK

Documentos de Trabajo. N.º 1726

2017

(*) I thank Máximo Camacho, Marcelle Chauvet, James D. Hamilton and Gabriel Pérez-Quirós, the editor and two anonymous referees for their helpful comments and suggestions. I also benefited from conversations with James Morley and Michael T. Owyang. Thanks to the seminar participants at the Bank of Canada, Bank of Mexico, Central Bank of Chile and University of California Riverside for helpful comments. Supplementary material of this paper can be found at the author’s webpage: https://sites.google.com/site/daniloleivaleon/media. The views expressed in this paper are those of the author and do not represent the views of the Banco de España.(**) [email protected]

Danilo Leiva-Leon (**)

BANCO DE ESPAÑA

MEASURING BUSINESS CYCLES INTRA-SYNCHRONIZATION

IN US: A REGIME-SWITCHING INTERDEPENDENCE FRAMEWORK (*)

The Working Paper Series seeks to disseminate original research in economics and fi nance. All papers have been anonymously refereed. By publishing these papers, the Banco de España aims to contribute to economic analysis and, in particular, to knowledge of the Spanish economy and its international environment.

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© BANCO DE ESPAÑA, Madrid, 2017

ISSN: 1579-8666 (on line)

Abstract

This paper proposes a Markov-switching framework to endogenously identify periods where

economies are more likely to (i) synchronously enter recessionary and expansionary phases,

and (ii) follow independent business cycles. The reliability of the framework is validated with

simulated data in Monte Carlo experiments. The framework is applied to assess the time-

varying intra-country synchronization in US. The main results report substantial changes

over time in the cyclical affi liation patterns of US states, and show that the more similar

the economic structures of states, the higher the correlation between their business cycles.

A synchronization-based network analysis discloses a change in the propagation pattern of

aggregate contractionary shocks across states, suggesting that the US has become more

internally synchronized since the early 1990s.

Keywords: business cycles, Markov-Switching, network analysis.

JEL classifi cation: E32, C32, C45.

Resumen

Este artículo propone un modelo econométrico de regímenes markovianos para identifi car

endógenamente períodos en los que las economías tienden a experimentar fases del ciclo

económico de manera sincronizada e independiente. La fi abilidad del modelo econométrico

se ha validado con datos simulados utilizando experimentos de Monte-carlo. El modelo se

aplica para identifi car cambios en la sincronización regional de Estados Unidos (EEUU).

Los resultados indican la presencia de cambios signifi cativos en los patrones cíclicos de

afi liación de los estados de EEUU, y muestran que, cuanto más similares son las estructuras

económicas de los estados, mayor es la correlación entre sus ciclos económicos.

Adicionalmente, un análisis de redes, basado en las medidas de sincronización estimadas,

revela un cambio en la propagación de choques contractivos a través de estados, lo que

sugiere que EEUU se ha sincronizado internamente con mayor intensidad desde principios

de los años noventa.

Palabras clave: ciclos económicos, cambios markovianos, análisis de redes.

Códigos JEL: E32, C32, C45.

BANCO DE ESPAÑA 7 DOCUMENTO DE TRABAJO N.º 1726

1 Introduction

Since Hamilton (1989), Markov-switching (MS) models have become a useful tool for policymakers and investors to construct inferences about the state of the economy (expansionary orrecessionary regimes), financial markets (high or low volatile regimes), monetary policy (activeor passive policy regimes), etc. Also, multivariate extensions of MS models have been usedto provide helpful insights about issues such as business cycles synchronization (Camacho andPerez-Quiros (2006)), business cycles and stock market volatility interdependence (Hamiltonand Lin (1996)), real activity and inflation cycles synchronization (Leiva-Leon (2014)), mone-tary and fiscal policy interaction (Davig and Leeper (2006)), among other types of relationships.In these studies, a key component of the analysis is the dependency relationship between theunderlying Markovian latent variables governing the model’s dynamics.

The modelling approaches of multivariate MS specifications can be sorted into two cate-gories. The first category includes studies where the relationship between the latent variablesis a priori defined. Hence, it is based on the researcher’s judgment, relying on four differentsettings (Hamilton and Lin (1996) and Anas et al. (2007)). The first refers to the case whereall series in the model are subject to a single latent variable (Krolzig (1997) and Sims and Zha(2006)). The second uses different latent variables which are modelled as totally independentMarkov chains (Smith and Summers (2005) and Chauvet and Senyuz (2008)). In the third,the dynamics of one latent variable precedes those of other latent variables (Hamilton andPerez-Quiros (1996) and Cakmakli et al. (2011)), allowing for a possibly different number oflags.1 Fourth, there is also the case of a general Markovian specification that involves the fulltransition probability matrix (Kim, Piger and Startz (2007)). However, it raises computationaldifficulties and is less straightforward to interpret as the number of series, states or lags, in-crease. Accordingly, the obtained regime inferences and final interpretations of the model’soutput may vary substantially depending on the approach chosen.

The second category focuses on making a posteriori assessments of the synchronizationbetween MS processes, providing “average” dependency relationship estimates. Works in thisline are Guha and Banerji (1998) and Artis et al. (2004), which focus on business cycles syn-chronization. The authors first estimate different MS univariate models and then computecross-correlations between the probabilities of being in recession as measures of synchroniza-tion.2 Phillips (1991) points out the two extreme cases presented in the literature: the case ofcomplete independence (two independent Markov processes are hidden in the bivariate specifi-cation) and the case of perfect synchronization (only one Markov process for both variables).Camacho and Perez-Quiros (2006) and Bengoechea et al. (2006) focus on assessing whether thelatent variables in multivariate models are either unsynchronized or perfectly synchronized bymodelling the data-generating process as a linear combination between the two cases. Leiva-Leon (2014) extends this approach to state-space representations, where the state vector isdriven by latent variables following dynamics that are modelled as linear combination betweenthe two polar cases. However, Leiva-Leon (2014) and previous related studies, assume that theweights assigned to each polar case, which are used to measure the synchronization between

the latent variables, are assumed to be constant over time.

1Another type of relationship, under a univariate framework, is presented in Bai and Wang (2011), wherethe state variable governing the mean of the process is conditional to the one governing the variance of thatprocess.

2However, as shown in Camacho and Perez-Quiros (2006), these approaches may lead to misleading results,since they are biased toward showing relatively low values of synchronization precisely for countries that exhibitsynchronized cycles. This suggests that a bivariate framework would provide a better characterization of pairwisesynchronization than two univariate models.


Despite the usefulness of the approaches used in the literature stream to deal with mul-tivariate MS models, they assume, or estimate, constant over time dependency relationshipsbetween the underlying latent variables governing the model’s dynamics. This assumptionmakes unfeasible assessments of endogenous changes in the structural relationship between thelatent variables. For example, in the case of business cycles synchronization, two economiesmay become more synchronized due to trade agreements, economic unions, etc. Therefore, theanalysis of changes in the structural relationship between the business cycles of theses economies(identified with the underlying latent variables) becomes crucial for the evaluation of specificpolicies.

Moreover, the study of business cycle synchronization is useful to assess the degree of ex-posure that a given economy has to its external environment. Previous works have used mul-tivariate MS models to study the synchronization of national economies (Smith and Summers(2005) and Camacho and Perez-Quiros (2006)), or regional economies (Owyang et al. (2005)and Hamilton and Owyang (2012)), providing synchronization patterns that are constant overtime. However, such degree of exposure may experience changes over time, which can be causedby a variety of factors, such as global recessionary shocks, global financial crises, etc. Therefore,changes in synchronization over time by using MS models can only be captured by splitting thesample into sub-periods. The problem with this approach is that its output relies on specificdate breaks, which sometimes may be controversial and might increase the risk of pretestingbias (Diebold, 2015). To the best of my knowledge, the time-varying relationship between thelatent variables of a MS model is an issue that still has not being studied from an endogenousperspective.

This paper proposes an approach to endogenously infer structural changes in the relation-ship between the latent variables governing multivariate MS models. For simplicity of thepresentation and without loss of generality, in the sequel, I focus on the case of business cyclessynchronization, however, the proposed framework can be applied to a wide range of appli-cations of multivariate MS models. The proposed framework endogenously identify regimeswhere two economies enter recessions and expansions synchronously, from regimes where theeconomies are unsynchronized and experience independent business cycle phases. In contrastto existing MS models in the literature, the filter of the proposed framework not only providesthe inferences associated to each latent variable, but it also provides simultaneous inferences onthe dependency relationship between the latent variables for each period of time. The modelis estimated by Gibbs sampling and its reliability is assessed with Monte Carlo experiments,suggesting it as a suitable approach to track changes in the synchronization of cycles.

Dynamic Factor Models have been widely used is assessing business cycles synchronizationby looking at the variability of an economy’s output growth explained by a “global component”,see Kose et al. (2012), Kose et al. (2003) for a constant parameter version, and Del Negro andOtrok (2008) for a time-varying parameter version. However, they provide no information onbilateral synchronizations, i.e. economy-specific business cycles pairwise interlinkages, whichis fundamental to study the dynamic propagation mechanism of business cycle shocks from adisaggregated perspective. The proposed framework provides time-varying pairwise synchro-nizations obtained from bivariate MS models that can be easily converted into measures ofdissimilarity, or business cycle distances. These distances can be used to assess changes inthe interdependence and clustering patterns experienced by a large set of economies by re-lying on network analysis. In such network, the economies take the interpretation of nodes,and the stochastic links between pairs of nodes is given by the estimated synchronicity, fullycharacterizing a business cycle network governed by Markovian dynamics.

The proposed framework is applied to investigate potential variations in the business cyclesinterdependence of U.S. states, and to assess the explanatory factor of the complex interactions


at the regional level, obtaining four main findings. First, the results report the existence of“interdependence cycles”, which are associated with recessions as identified by the NationalBureau of Economic Research (NBER). Such cycles are defined as periods characterized bylow cyclical heterogeneity across U.S. states, experienced during the recessionary and recoveryphases, followed by longer periods of high cyclical heterogeneity, which occurs during the phasesof stable growth. Second, there are substantial variations in the grouping pattern of states overtime, going from a scheme characterized by several clusters of states to a core and peripherystructure, composed of highly and lowly synchronized states, respectively. Third, the networkanalysis documents a change in the propagation pattern of contractionary shocks across states.Until the 1990s, recessions were characterized by the spread of shocks mainly across a fewbig states in terms of their share of GDP. Since that time, recessionary shocks have beenmore uniformly spread across all states, suggesting that regions of the U.S. economy havebecome more interdependent over the past two decades. Fourth, the main factor driving USintra-synchronization is the similarity of the economic structure across states, the more similarthe structures, the more similar the responsiveness to shocks, and therefore, the higher thecorrelation between their business cycles. Also, more similar states in terms of householdwealth tend to experience higher business cycles synchronization.

The paper is structured as follows. Section 2 presents the proposed time-varying synchro-nization approach. Section 3 reports the Monte Carlo simulation results. Section 4 analyzesthe dynamic synchronization of business cycle phases in U.S. states, relying on bivariate, mul-tivariate and network analyses. Finally, Section 5 concludes.

2 The Model

Let yi,t be the growth rate of an economic activity index of economy i, which can be modelled asa function of a latent or unobserved state variable (Si,t) that indicates whether the economy is ina recessionary or expansionary regime, an idiosyncratic component, εi,t, and a set of additionalparameters, θi. Accordingly, for i = a, b,

ya,t = f(Sa,t, εa,t, θa) (1)

yb,t = f(Sb,t, εb,t, θb). (2)

The goal of this section is to provide assessments on the synchronization between Sa,t and Sb,t

for each period of time; that is,

sync(Sa,t, Sb,t) = Pr(Sa,t = Sb,t), for t = 1, ..., T. (3)

Following Owyang et al. (2005) and Hamilton and Owyang (2012), who rely on AR(0)MS specification, the following tractable bivariate two-state Markov-switching specification isconsidered: [

ya,tyb,t

]=

[μa,0 + μa,1Sa,t

μb,0 + μb,1Sb,t

]+

[εa,tεb,t

], (4)

where the innovations εt = (εa,t, εb,t)′ are assumed to have a variance-covariance matrix that

experiences changes of regimes, that is, εt ∼ N(0,Σt), where

Σt = Σ0(1−Gt) + Σ1Gt, (5)

and Gt denotes an unobserved state variable that accounts for volatility regimes, and it isassumed to be independent from Sa,t and Sb,t.

It is worth noting that the results derived in this section can be straightforwardly extendedto specifications including lags in the dynamics. However, Camacho and Perez-Quiros (2007)


show that positive autocorrelation in macroeconomic time series can be better captured byshifts between business cycle states rather than by the standard autoregressive coefficients.3

This result agrees with Albert and Chib (1993), who show that the posterior distribution ofautoregressive parameters tend to be centered at zero when modelling US output growth withregime-switching models.

When Sk,t = 0, the state variable Sk,t indicates that ykt is in regime 0 with a mean equalto μk,0. When Sk,t = 1, ykt is in regime 1 with a mean equal to μk,0 + μk,1, for k = a, b.Moreover, Sa,t and Sb,t evolve according to irreducible two-state Markov chains, whose transitionprobabilities are given by

Pr(Sk,t = j|Sk,t−1 = i) = pk,ij, for ik, jk = 0, 1 and k = a, b. (6)

Analogously, when Gt = 0, the state variable Gt indicates that εt is in regime 0 with a variance-covariance matrix Σ0. When Gt = 1, εt is in regime 1 with variance-covariance matrix Σ1. Thestate variable Gt follows a two-state Markov chain with transition probabilities given by

Pr(Gt = jg|Gt = jg) = pg,ij, for ig, jg = 0, 1 (7)

To characterize the dynamics of yt = [ya,t, yb,t]′, the information contained in Sa,t, Sb,t, and

Gt, can be summarized in the state variable, Sab,t, which accounts for the possible combinationsthat the vector μt = [μa,0 + μa,1Sa,t, μb,0 + μb,1Sb,t]

′ and Σt could take through the differentregimes:

Sab,t =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1, if Sa,t = 0, Sb,t = 0, Gt = 02, if Sa,t = 0, Sb,t = 1, Gt = 03, if Sa,t = 1, Sb,t = 0, Gt = 04, if Sa,t = 1, Sb,t = 1, Gt = 05, if Sa,t = 0, Sb,t = 0, Gt = 16, if Sa,t = 0, Sb,t = 1, Gt = 17, if Sa,t = 1, Sb,t = 0, Gt = 18, if Sa,t = 1, Sb,t = 1, Gt = 1

. (8)

Similar to Harding and Pagan (2006), the objective of the proposed model is to differentiateregimes where the phases of ya,t and yb,t are unsynchronized, implying that Sa,t and Sb,t followindependent dynamics; that is,

Pr(Sa,t = ja, Sb,t = jb, Gt = jg) = Pr(Sa,t = ja) Pr(Sb,t = jb) Pr(Gt = jg), (9)

from regimes where the phases of ya,t and yb,t are fully synchronized, entering expansions andrecessions synchronously, implying that Sa,t = Sb,t = St; that is,

Pr(Sa,t = ja, Sb,t = jb, Gt = jg) = Pr(St = j) Pr(Gt = jg). (10)

In order to do so, I introduce into the framework another latent variable, Vt, that takes thevalue of 1 if business cycle phases are in a synchronized regime, and the value of 0 if they areunder an unsynchronized regime at time t; that is,

Vt =

{0 if Sa,t and Sb,t are unsynchronized1 if Sa,t and Sb,t are synchronized.

(11)

The latent variable Vt also evolves according to an irreducible two-state Markov chain whosetransition probabilities are given by

Pr(Vt = jv|Vt−1 = iv) = pv,ij, for iv, jv = 0, 1. (12)

3This finding is in line with the results obtained by Kim, Morley and Piger (2005) and Morley and Piger(2006) who focus on assessing the importance of the “third phase” in the business cycle, and find that there isno need for autoregressive coefficients in the growth rates once the nonlinearities are correctly specified.


The advantage of introducing Vt, rather than analyzing the general Markovian specificationwith the full transition probability matrix, as in Sims et al. (2008), is that all the informationabout the dependency relationship between the latent variables remains summarized in a singlevariable, Vt, providing an easy-to-interpret way of assessing synchronization changes. It isalso able to provide information about the expected duration of regimes where economies aresynchronized or unsynchronized based on their associated transition probabilities. Notice thatthe analysis in this paper focuses on dependency, not on correlations, since the objective is todetermine if two economies are either synchronized or unsynchronized.

Accordingly, there is an enlargement of the set of regimes in Equation (8), which remainsfully characterized by the latent variable S∗

ab,t, that simultaneously collects information regard-ing joint dynamics, individual dynamics and their dependency relationship over time:

S∗ab,t =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1, if Sa,t = 0, Sb,t = 0, Gt = 0, Vt = 02, if Sa,t = 0, Sb,t = 1, Gt = 0, Vt = 03, if Sa,t = 1, Sb,t = 0, Gt = 0, Vt = 04, if Sa,t = 1, Sb,t = 1, Gt = 0, Vt = 05, if Sa,t = 0, Sb,t = 0, Gt = 1, Vt = 06, if Sa,t = 0, Sb,t = 1, Gt = 1, Vt = 07, if Sa,t = 1, Sb,t = 0, Gt = 1, Vt = 08, if Sa,t = 1, Sb,t = 1, Gt = 1, Vt = 09, if Sa,t = 0, Sb,t = 0, Gt = 0, Vt = 110, if Sa,t = 0, Sb,t = 1, Gt = 0, Vt = 111, if Sa,t = 1, Sb,t = 0, Gt = 0, Vt = 112, if Sa,t = 1, Sb,t = 1, Gt = 0, Vt = 113, if Sa,t = 0, Sb,t = 0, Gt = 1, Vt = 114, if Sa,t = 0, Sb,t = 1, Gt = 1, Vt = 115, if Sa,t = 1, Sb,t = 0, Gt = 1, Vt = 116, if Sa,t = 1, Sb,t = 1, Gt = 1, Vt = 1

. (13)

Inferences on the latent variable S∗ab,t, can be computed by conditioning on Vt

4:

Pr(S∗ab,t = j∗ab) = Pr(Sa,t = ja, Sb,t = jb, Gt = jg, Vt = jv)

= Pr(Sa,t = ja, Sb,t = jb, Gt = jg|Vt = jv) Pr(Vt = jv), (14)

where Pr(Sa,t = ja, Sb,t = jb, Gt = jg|Vt = jv) indicates the inferences on the dynamics of Sab,t,conditional on total independence if Vt = 0, or conditional on full dependence if Vt = 1. In theformer case, the joint probability of S∗

ab,t is given by

Pr(Sa,t = ja, Sb,t = jb, Gt = jg, Vt = 0) = Pr(Sa,t = ja, Sb,t = jb, Gt = jg|Vt = 0)Pr(Vt = 0)

= Pr(Sa,t = ja) Pr(Sb,t = jb) Pr(Gt = jg) Pr(Vt = 0), (15)

while, in the latter case, it is given by

Pr(Sa,t = ja, Sb,t = jb, Gt = jg, Vt = 1) = Pr(Sa,t = ja, Sb,t = jb, Gt = jg|Vt = 1)Pr(Vt = 1)

= Pr(St = j) Pr(Gt = jg) Pr(Vt = 1). (16)

4Notice that states 10, 11, 14 and 15 in Equation (13) are truncated to zero by construction, since the two statevariables cannot be in different states if they are perfectly synchronized, i.e., Pr(Sa,t = ja, Sb,t = jb|Vt = 1) = 0for any ja �= jb.


Vt; that is,

Pr(Sa,t = ja, Sb,t = jb, Gt = jg) = Pr(Vt = 1)Pr(St = j) Pr(Gt = jg) +

(1− Pr(Vt = 1)) Pr(Sa,t = ja) Pr(Sb,t = jb) Pr(Gt = jg), (17)

which implies that the joint dynamics of Sa,t, Sb,t, and Gt remain characterized by a weightedaverage between the extreme dependent and independent cases, where the weights assigned toeach of them are endogenously determined by

Pr(Vt = 1) = δabt . (18)

Therefore, from now on, the term δabt will be referred to as the dynamic synchronicitybetween Sa,t and Sb,t.

2.1 Filtering Algorithm

This section develops an extension of the Hamilton (1994) algorithm to estimate the modeldescribed in Equations (4) and (17). The algorithm is composed of two unified steps. In thefirst one, the goal is the computation of the likelihoods, while in the second, the goal is theprediction and updating of probabilities.

STEP 1: The parameters of the model are assumed to be known for the moment and arecollected in the vector

θ = (μa,0, μa,1, μb,0, μb,1,Σ0,Σ1, pa,00, pa,11, pb,00, pb,11, p00, p11, pv,00, pv,11, pg,00, pg,11)′. (19)

The conditional joint density corresponding to the state variable that fully characterizes themodel’s dynamics, S∗

ab,t, can be expressed as a function of its components,

f(yt, S∗ab,t = j∗ab|ψt−1; θ) = f(yt, Sa,t = ja, Sb,t = jb, Gt = jg, Vt = jv|ψt−1; θ), (20)

which is the product of the density, conditional on the realization of the set of regimes timesthe probability of occurrence of such realizations,

f(yt, Sa,t = ja, Sb,t = jb, Gt = jg, Vt = jv|ψt−1; θ) = f(yt|Sa,t = ja, Sb,t = jb, Gt = jg, Vt = jv, ψt−1; θ)×Pr(Sa,t = ja, Sb,t = jb, Gt = jg, Vt = jv|ψt−1; θ). (21)

The joint probability of Sa,t = ja, Sb,t = jb, Gt = jg and Vt = jv is obtained by using conditionalprobabilities,

Pr(Sa,t = ja, Sb,t = jb, Gt = jg, Vt = jv|ψt−1; θ) = Pr(Sa,t = ja, Sb,t = jb, Gt = jg|Vt = jv, ψt−1; θ)×Pr(Vt = jv|ψt−1; θ), (22)

where the term Pr(Sa,t = ja, Sb,t = jb, Gt = jg|Vt = jv, ψt−1; θ) is fully characterized with theresults derived in Equations (15) and (16). Thus, Equation (22) remains a function of onlyPr(Sk,t = jk|ψt−1; θ) for k = a, b, Pr(Gt = jg|ψt−1; θ), Pr(Vt = jv|ψt−1; θ) and Pr(St = j|ψt−1; θ).The steady state or ergodic probabilities can be used as starting values to initialize the filter.

Therefore, inferences on the state variable Sab,t, in Equation (8), after accounting for synchro-nization, can be easily recovered by integrating Pr(Sa,t = ja, Sb,t = jb, Gt = jg, Vt = jv) through


are obtained asIn order to make inferences on the evolution of single-state variables, the marginal densities

f(yt, Sa,t = ja|ψt−1; θ) =1∑

jb=0

1∑jg=0

1∑jv=0

f(yt, Sa,t = ja, Sb,t = jb, Gt = jg, Vt = jv|ψt−1; θ),(23)

f(yt, Sb,t = jb|ψt−1; θ) =1∑

ja=0

1∑jg=0

1∑jv=0


f(yt, Gt = jg|ψt−1; θ) =1∑

ja=0

1∑jb=0

1∑jv=0


f(yt, Vt = jv|ψt−1; θ) =1∑

ja=0

1∑jb=0

1∑jg=0

f(yt, Sa,t = ja, Sb,t = jb, Gt = jg, Vt = jv|ψt−1; θ).(26)

The marginal density associated the state variable St requires a special treatment. When it isassumed that the model’s dynamics are governed by only one state variable, i.e., Sa,t = Sb,t = St,the density in Equation (20) collapses to f †(yt, St = j|ψt−1; θ), where

Accordingly, the density of yt, conditional on the past observables, is given by

f †(yt, St = 0|ψt−1; θ) =1∑

jg=0

f(yt, Sa,t = 0, Sb,t = 0, Gt = jg, Vt = 1|ψt−1; θ), (27)

f †(yt, St = 1|ψt−1; θ) =1∑

jg=0

f(yt, Sa,t = 1, Sb,t = 1, Gt = jg, Vt = 1|ψt−1; θ), (28)

f †(yt|ψt−1; θ) =1∑

j=0

f †(yt, St = j|ψt−1; θ). (30)

STEP 2: Once yt is observed at the end of time t, the prediction probabilities Pr(Sk,t =jk|ψt−1; θ) for k = a, b, Pr(Gt = jg|ψt−1; θ), Pr(Vt = jv|ψt−1; θ) and Pr(St = j|ψt−1; θ) can beupdated:

f(yt|ψt−1; θ) =1∑

ja=0

1∑jb=0

1∑jg=0

1∑jv=0

f(yt, Sa,t = ja, Sb,t = jb, Gt = jg, Vt = jv|ψt−1; θ), (29)

and under the assumption that Sa,t = Sb,t = St, it is given by

Pr(Sa,t = ja|ψt; θ) =f(yt, Sa,t = ja|ψt−1; θ)

f(yt|ψt−1; θ)(31)

Pr(Sb,t = jb|ψt; θ) =f(yt, Sb,t = jb|ψt−1; θ)

f(yt|ψt−1; θ)(32)

Pr(Gt = jg|ψt; θ) =f(yt, Gt = jg|ψt−1; θ)

f(yt|ψt−1; θ)(33)

Pr(Vt = l|ψt; θ) =f(yt, Vt = l|ψt−1; θ)

f(yt|ψt−1; θ)(34)

Pr(St = j|ψt; θ) =f †(yt, St = j|ψt−1; θ)

f †(yt|ψt−1; θ)(35)


and Vt, respectively:

Pr(Sk,t+1 = jk|ψt; θ) =1∑

ik=0

Pr(Sk,t+1 = jk, Sk,t = ik|ψt; θ)

=1∑

ik=0

Pr(Sk,t+1 = jk|Sk,t = ik) Pr(Sk,t = ik|ψt; θ), for k = a, b (36)

Pr(Gt+1 = jg|ψt; θ) =1∑

ig=0

Pr(Gt+1 = jg, Gt = ig|ψt; θ)

=1∑

ig=0

Pr(Gt+1 = jg|Gt = ig) Pr(Gt = ig|ψt; θ) (37)

Pr(Vt+1 = jv|ψt; θ) =1∑

i=0

Pr(Vt+1 = jv, Vt = iv|ψt; θ)

=1∑

i=0

Pr(Vt+1 = jv|Vt = iv) Pr(Vt = iv|ψt; θ) (38)

Finally, the above forecasted probabilities are used to predict inferences on the realizationsof S∗

ab,t+1, relying on Equation (22):

Pr(Sa,t+1 = ja, Sb,t+1 = jb, Gt+1 = jg, Vt+1 = jv|ψt; θ) =

Pr(Sa,t+1 = ja, Sb,t+1 = jb, Gt+1 = jg|Vt+1 = jv, ψt; θ)× Pr(Vt+1 = jv|ψt; θ), (40)

where Equation (40) remains a function of Pr(Sk,t+1 = jk|ψt; θ) for k = a, b, Pr(Gt+1 = jg|ψt; θ),Pr(Vt+1 = jv|ψt; θ) and Pr(St+1 = j|ψt; θ).

By iterating these two steps for t = 1, 2, . . . , T , the algorithm provides simultaneous infer-ences on Sa,t, Sb,t, Gt, and their dynamic synchronicity δabt between Sa,t and Sb,t as defined inEquation (18).

Regarding the estimation of the parameters, notice that, as the number of possible statesincreases, the likelihood function could be characterized by several local maximums, causingstrong convergence problems in performing maximum likelihood estimations, as shown in Boldin(1996). Hence, given the high number of combinations of states through which the likelihoodis conditioned in Equation (29), the set of parameters θ along with the inferences on the statevariables are estimated by using Bayesian methods. Specifically, a multivariate version of theapproach in Kim and Nelson (1999), which applies Gibbs sampling procedures, is used. Theestimation method is explained in detail in Appendix A.

Pr(St+1 = j|ψt; θ) =1∑

i=0

Pr(St+1 = j, St = i|ψt; θ)

=1∑

i=0

Pr(St+1 = j|St = i) Pr(St = i|ψt; θ) (39)

Forecasts of the updated probabilities in Equations (31) to (35) are done by using the corre-sponding transition probabilities pa,ij, pb,ij, pg,ij, pij, pv,ij, in the vector θ, for Sa,t, Sb,t, Gt, St


consists of two steps. First, the generation of two stochastic processes subject to regime switch-ing that experience one or more synchronization changes. Second, by letting the econometricianobserve only the generated data, but not the data-generating process, the proposed filter inSection 2.1 along with the Gibbs sampler, are applied to obtain estimates of the model’s pa-rameters, probabilities of recession for each economy, and, more importantly, the inferenceson synchronization changes. I then address how well the parameter estimates and inferencesmatch the real ones.

Given a sample of size T , the data generating process consists of generate a two-state first-order Markovian process, Gt, with transition probability matrix

3 Simulation Study

In order to validate the reliability of the proposed approach to assess changes in the synchro-nization of business cycle phases, I rely on the use of Monte Carlo experiments. Each simulation

Given two variance-covariance matrices, Σ∗0 and Σ∗

1, generate the innovations et = [ea,t, eb,t]from a N(0,Σ∗

t ), whereΣ∗

t = Σ∗0(1−Gt) + Σ∗

1Gt. (42)

Next, generate a Markovian process, Sa,t, with a transition probability matrix,

P ∗g =

(p∗g,00 1− p∗g,11

1− p∗g,00 p∗g,11

). (41)

Then, given a vector of means μ∗a = [μ∗

a,0, μ∗a,1]

′, generate a process yIa,t as follows:

yIa,t = μ∗a,0 + μ∗

a,1Sa,t + ea,t, (44)

and given a vector of means μ∗b = [μ∗

b,0, μ∗b,1]

′, and transition probabilities p∗b,00 and p∗b,11, thesame procedure is repeated to independently generate

yIb,t = μ∗b,0 + μ∗

b,1Sb,t + eb,t, (45)

where Sb,t is a first-order Markovian process. Next, another Markovian process, St, is generatedby using the transition matrix

P ∗a =

(p∗a,00 1− p∗a,11

1− p∗a,00 p∗a,11

). (43)

P ∗ab =

(p∗00 1− p∗11

1− p∗00 p∗11

). (46)

Then, given the two vectors of means μ∗a and μ∗

b , generate jointly

[yDa,tyDb,t

]=

[μ∗a,0 + μ∗

a,1St

μ∗b,0 + μ∗

b,1St

]+

[ea,teb,t

]. (47)

The information generated so far can be collected in two vectors, one in which two stochasticprocesses are driven by two Markov-switching variables independent from each other, yIt =[yIa,t, y

Ib,t]

′, and the other where two stochastic processes are governed by only one Markov-switching dynamic, yDt = [yDa,t, y

Db,t]

′.The premise of this paper is that, during some regimes, the output growth of two economies

can follow dynamics similar to those in yDt , while during other regimes, things can change inone, or both, of the economies, leading their joint dynamics to behave in the same way as thosein yIt , following independent patterns. To mimic this situation, I start analyzing the simplestcase in which there is just one synchronization change in a sample of size T , occurring at time


τ , with 1 < τ < T .5 Then, I let yt = [ya,t, yb,t]′ be the observed output growth of two economies,

which comes from the following unobserved data generating process:

yt =

{yDt , for t = 1, . . . , τ

yIt , for t = τ + 1, . . . , T, (48)

which can be alternatively expressed as

yt = yDt Vt + (1− Vt)yIt , (49)

where Vt is an indicator variable of synchronization, whose dynamics are described by

{Vt}T1 =

[1τ

0T−τ

], (50)

with 1τ being a vector, with entries equal to one, of size τ , and 0T−τ a zero vector of sizeT − τ . The case of one synchronization change can be easily extended to mimic the case ofZ synchronization changes, occurred at τ1, τ2, . . . , τZ , with 1 < τ1 < τ2 < . . . < τZ < T , justby appropriately modifying the dynamics in {Vt}T1 . These experiments are evaluated underZ = 6 different scenarios. Each scenario corresponds to z changes in synchronization, forz = 1, 2, 3, 4, 5, and the last case considers a random number of synchronization changes, i.e.,unlike predefining the dynamics of Vt as in Equation (50), it is modelled as a first-order Markovchain with transition probabilities p∗V,00 and p∗V,11, i.e. z = f(Vt).

In addition, I study the performance of the proposed approach under a scenario where theassumption that Sa,t and Sb,t are either perfectly correlated or totally independent is relaxed.Accordingly, it is assume that the state variables Sa,t and Sb,t are imperfectly correlated. Inparticular, I generate a four-state Markovian process, S̄ab,t = {1, 2, 3, 4}, with its correspondingfull 4 × 4 transition probability matrix, Q. Based on the realizations of S̄ab,t, I generate thevector St according to:

St =

⎧⎪⎪⎨⎪⎪⎩

(0, 0), if S̄ab,t = 1(0, 1), if S̄ab,t = 2(1, 0), if S̄ab,t = 3(1, 1), if S̄ab,t = 4

(51)

Then, given the matrix of means μ =[μ∗a, μ

∗b ], generate observed data, yICt , from state variables

experiencing imperfect correlation, that is,

yICt = μ� St + et, (52)

where � represents the Hadamard product. The entries of the matrix Q are set to produce agiven level of correlation, δ, between Sa,t and Sb,t. Specifically, three levels of correlation areevaluated, high, medium, and low, with δ = 0.7, 0.5, 0.2, respectively.6 Therefore, the dataobserved by the econometrician is produced following the data generating process:

yt =

⎧⎨⎩

yDt , for t = 1, . . . , T/3yICt , for t = T/3 + 1, . . . , T (2/3)yIt ,, for t = T (2/3) + 1, . . . , T

5The selection of τ is based on a random draw u, generated from a uniform distribution U [0, 1], i.e., τ̂ = uT ,then τ̂ is rounded to the nearest integer number to obtain τ . Also, the use of draws of τ equal to the boundaries,i.e., 1 or T , is avoided.

6For each level of correlation, δ, the matrix Q is calibrated such that ρ̄ � δ, where ρ̄ is the average correlationbetween S̄a,t and S̄b,t over 10, 000 simulations of S̄ab,t, and the state variables are defined as

S̄a,t =

{0, if S̄ab,t = 3 or S̄ab,t = 4

1, Otherwise, S̄b,t =

{0, if S̄ab,t = 2 or S̄ab,t = 4

1, Otherwise


Since the data-generating process and parameters are unknown by the econometrician, theGibbs sampler is used to estimate the model’s parameters, the probabilities of recession for eacheconomy, the probability of highly volatile output, and, more importantly, inferences on thedynamics of Vt, by relying on the filtering algorithm proposed in Section 2.1. The criterion usedto assess the performance of the regime inferences and the synchronization is the QuadraticProbability Score (QPS), defined as

QPS(Ξ) =1

T

T∑t=1

(Ξ− Pr(Ξ = 1|ψT ))2, for Ξ = Sa,t, Sb,t, Gt, Vt. (53)

To illustrate the filtering and estimation strategy’s performance, Figure 1 plots one simu-lation for the cases in which there is one, two and three synchronization changes in a sampleof 400 periods, i.e., for z = 1, 2, 3, with T = 400.7 For each case, the top charts plot the twoobserved time series, ya,t and yb,t, generated with the parameter values in Table 1 and by usingEquation (49), along with the unobserved dynamics of Vt. Both time series show strong coher-ence in phases when Vt = 1, and the opposite occurs when Vt = 0. The second row of chartsplot the computed inferences on the synchronization changes, i.e., Pr(Vt = 1), along with thetrue dynamics of Vt, showing their close relation in all three cases and providing insight intothe satisfactory performance of the proposed framework for assessing synchronization changes.The third row of charts plot the two observed series along with the unobserved dynamics ofthe state variable Gt. Notice that ya,t and yb,t experience more volatile fluctuations duringperiods where Gt = 1, and less volatile dynamics when Gt = 0. Finally, the fourth row ofcharts plot the computed inferences on regimes of high volatility, i.e., Pr(Gt = 1), along withthe true dynamics of Gt, showing that the model is also able to perform accurate inferences ofthe volatility regimes.

The parameters used in the simulations exercises are specified in Table 1 and the experi-ments associated to each scenario are replicated M = 1, 000 times. The results of the MonteCarlo simulations are reported in Table 2, showing the average over the M replications of eachestimated parameter

θ∗z =1

M

M∑m=1

θ∗(m)z , (54)

where θ∗(m)z corresponds to the vector of parameters, as defined in Equation (19), associated

to the m-th replica and the z-th case. All parameter estimates appear to be unbiased for thedifferent values of z and δ. Notice that the stochastic process with the highest difference ofthe within-regime means, in this case yb,t, shows more accurate estimates, meaning that higherdifferences provide a better identification of the phases of the business cycles. Regarding theperformance of the regime inferences, Table 3 reports the averages over the M replications withthe QPS associated with the state variables Sa,t, Sb,t and Vt, which can be interpreted as theaverage over the M replications of the squared deviation from the generated business cycles:

QPS(Ξ)ζ =1

M

M∑m=1

QPS(Ξ)(m)ζ , for Ξ = Sa,t, Sb,t, Gt, Vt, and for ζ = z, δ, (55)

where QPS(Ξ)(m)ζ , as defined in Equation (53), corresponds to the m-th replica and the ζ-th

scenario, that corresponds to a specific value of z or δ. The results indicate that, althoughinferences on the state variables in general present high precision, the ones associated with thetime series with the highest difference of the within-regime means, yb,t, are, in general, the mostaccurate.

7We choose this sample size since it is close to the one used in the empirical application of Section 4.


The precision of the inferences on synchronization changes decreases as the number ofchanges, k, increases. This feature can also be observed by looking at the histograms of the Mreplications plotted in Figure 1 of Appendix B, in particular, the last column of charts, wherethe distribution of QPS(Vt)

(m)ζ is shown. However, it is natural to think of synchronization

changes as events that do not occur as often as the business cycle phases of an economy. Theymay require longer periods of time to take place, since they originate from changes in thestructural relationships among economies. This suggests that the proposed model is suitablefor accurately inferring synchronization changes of business cycle phases. Moreover, the modelis able to appropriately characterize the underlying level of imperfect correlation between Sa,t

and Sb,t, as can be seen in the last row of Table 3, for lowly, moderately and highly correlatedstate variables, respectively.

4 Monitoring U.S. States Business Cycles Synchroniza-

tion

The most recent global financial crisis has stimulated interest in the study of the sources andpropagation of contractionary episodes, calling for a more careful look at the disaggregation ofbusiness cycles in order to assess the mechanisms underlying economic fluctuations.

On the one hand, recent work by Acemoglu et al. (2012), which relies on network analysis,finds that sectoral interconnections capture the possibility of “cascade effects,” whereby pro-ductivity shocks to a sector propagate not only to its immediate downstream customers, butalso to the rest of the economy. On the other hand, two recent papers have shown interestingfeatures of economic activity synchronization when the business cycle is disaggregated at theregional level. In the first, Owyang et al. (2005) investigate the evolution of the individual busi-ness cycle phases of U.S. states. By following a univariate approach, the authors find that U.S.states differ significantly in the timing of switches between expansions and recessions, and alsodiffer in the extent to which phases in state business cycles are synchronous with those of thenational economy. In the second paper, Hamilton and Owyang (2012) use a unified frameworkto go through the propagation of regional recessions in the United States, using a multivariateapproach that focuses on clustering the states that share similar business cycle characteristics.They find that differences across states appear to be a matter of timing and that they can begrouped into three clusters, with some entering recession or recovering before others. Althoughthese previous studies provide useful insights about the overall synchronization pattern in agiven sample period, they are not able to detect changes in patterns occurring in these timespans.

This study intends to unify both concepts: first, the dynamic synchronization of pairwisecycles, by using the framework proposed in Section 2; and second, the dynamic interdependenceamong all U.S. states, by relying on network analysis, in order to assess the presence and thenature of potential changes in the regional propagation of contractionary shocks. For thispurpose, I use data on U.S. states coincident indexes, proposed in Crone and Matthews(2005)and provided by the Federal Reserve Bank of Philadelphia, as monthly indicators of the overalleconomic activity at the state level. The sample spans from August 1979, when the data for allthe states started to be reported, until February 2016 (Alaska and Hawaii are excluded as inHamilton and Owyang (2012)). The Chicago Fed National Activity Index (CFNAI) is used as amonthly measure of the U.S. national business cycle. All these indexes of real economic activity,for each state and for the entire United States, have been constructed based on the principleof co-movement among industrial production, employment, sales and income measures.


4.1 Bivariate Analysis

The analysis for 48 states plus the United States as a whole requires the modelling of the C492 =

1, 176 pairwise comparisons. To assess the performance of the proposed Markov-switchingsynchronization model, two selected examples are analyzed in detail.8

The first example focuses on the case of two states that have a high share of national GDP:New York (7.68%) and Texas (7.95%). Table 4 reports the Bayesian estimates for the New Yorkvs. Texas model, showing almost zero growth rates when St = 0 and positive growth when St =1, for both states. It is worth highlighting the estimates of the transition probabilities associatedwith the state variable that measures synchronization, Vt. The probability of remaining in alow synchronization regime, 0.97, is slightly higher than the probability of remaining in a lowsynchronization regime, 0.94. This result is corroborated in the first three rows of Chart A ofFigure 2, which plots (i) the probabilities of recession for New York and (ii) for Texas alongwith (iii) the corresponding time-varying synchronization, δNY,TX

t , as defined in Equation (18).As can be seen, from the 1980s to the mid-1990s, these states experienced recessions at differenttimes. This is reflected in the low values of the synchronicity. However, since the mid-1990s,both economies have been experiencing the same recession chronology, which is consistent withthe increase in the synchronicity observed after the mid-1990s. Also, the model controls forpotential changes in the variance-covariance matrix of innovations by inferring the probabilityof high volatile real activity for the two states, shown in the fourth row of Chart A of Figure 2.These probabilities indicate a high volatility regime during the pre-Great Moderation periodand also during the Great Recession.

The second example analyzes the case of two states with different shares of GDP: the statewith the highest, California (13.34%); and the state with the lowest, Vermont (0.18%). Table5 presents the Bayesian parameter estimates of the model. Unlike the previous example, theprobability California vs. Vermont remain highly synchronized, 0.99, is higher than the proba-bility of remaining unsynchronized, 0.87. This is also illustrated in Chart B of Figure 2, whichshows that, in general, both states have experienced the same business cycle chronology, enter-ing recessions and expansions synchronously, with the exception of one period. Specifically, in1989, Vermont entered a recessionary phase, while California was still growing until mid-1990,when it also started to experience a recession. However, at the beginning of 1992, Vermontstarted an expansionary phase, while California remained in recession until 1994. These desyn-chronicities are reflected in the downturn of the dynamic synchronization, δCA,V T

t , during thatperiod. Also, Chart B of Figure 2 plots the probability of high volatility regime, showing highvalues during the 1990 recession and during the Great Recession.

Considerable heterogeneity was found in the dynamics of the estimated time-varying syn-chronizations, finding cases involving significant changes, and cases where the synchronizationwas almost constant, at low or high levels. Although the proposed framework can provideinformation on the synchronization between any pair of states for any given period of time,other ways to summarize the information are needed, since policy-makers are interested in the“big picture” of the overall regional synchronization path.

4.2 Multivariate Analysis

As suggested by Tim (2002) and Camacho et al. (2006), the multi-dimensional scaling (MDS)method is a helpful tool for identifying cyclical affiliations between economies, since it seeksto find a low-dimensional coordinate system to represent n-dimensional objects and create amap of lower dimension (k). Traditionally, studies use as input for this method a symmetricmatrix, Γ, that summarizes the cyclical distances between economies for a given time span.

8The results for the other 1,174 cases are available from the author upon request.


Each entry γij of the matrix assigns a value characterizing the distance between economies iand j. The output of the MDS consists of one map showing the general picture for all thecyclical affiliations.

The dynamic synchronization measures obtained in the bivariate analysis, 0 ≤ δijt ≤ 1, canbe easily converted into desynchronization measures, γij

t = 1 − δijt . Accordingly, γijt can be

interpreted as cyclical distances, allowing the construction of the dissimilarity matrix Γ, foreach time period:

Γt =

⎛⎜⎜⎜⎜⎜⎝

1 γ12t γ13

t . . . γ1nt

γ21t 1 γ23

t . . . γ2nt

γ31t γ32

t 1 . . . γ3nt

......

.... . .

...γn1t γn2

t γn3t . . . 1

⎞⎟⎟⎟⎟⎟⎠

, (56)

which provides the possibility of assessing changes in the general picture of all cyclical affiliationsof U.S. states.

In a recent work on MDS, Xu et al. (2012) propose a way to deal with MDS in a dynamicfashion, where the dimensional coordinates of the projection of any two objects, i and j, arecomputed by minimizing the stress function,

minγ̃ijt=

n∑i=1

n∑j=1

(γijt − γ̃ij

t )2

∑i,i(γ

ijt )

2+ β

n∑i=1

γ̃it|t−1, (57)

where

γ̃ijt = (||zi,t − zj,t||2)1/2 (58)

γ̃it|t−1 = (||zi,t − zi,t−1||2)1/2, (59)

zi,t and zj,t are the k-dimensional projection of the objects i and j, and β is a temporal regu-larization parameter that serves to zoom in or zoom out changes between frames at t and att+1, always keeping the same dynamics independent of its value. In principle, β can be simplyset up to 1; however, since the data in Γt belong to the unit interval, for a more adequatevisual perception of the transitions between frames it is set up to 0.05. The output of theminimization in Equation (57) provides a two-dimensional representation of Γt.

The synchronization maps of U.S. states for the first month of the last four recessions areplotted in the charts of Figure 3. Each point in the charts represents a state, and the middlepoint refers to the United States as a whole. The closeness between two points in the plane refersto their degree of synchronicity, i.e., the closer the points are, the greater their synchronization.The figure corroborates the premise in the introduction of this paper about the existence ofsignificant changes in the grouping pattern among regional economies over time.

Specifically, Chart A plots the scenario for the 1981 recession, showing a few clusters ofstates experiencing similar business cycles phases. Notice that only a few states, such as SouthCarolina and Washington, located inside the first concentric circle of the chart, were highlysynchronized with the national business cycle. Instead, most of the states were located inthe second concentric circle, experiencing moderated synchronization. The remaining states,located in the third concentric circle, such as Florida, Colorado, Texas, North Dakota, WestVirginia, among others, were lowly synchronized between each other and with respect to thenational cycle, following mostly independent patterns. Chart B presents the situation for the1990 recession, showing a slightly different clustering pattern between states, but keeping threegroups, highly, moderately and lowly synchronized states with the national cycle. Charts Cand D present the scenarios for the 2001 and 2007 recessions, in the left and right corner,


respectively. Both charts indicate a stronger synchronization pattern between states and withrespect to the national business cycle, characterized by a core (composed of states highly insync) and periphery (composed of independent states) structure. In both periods, the firstconcentric circle contains a large number of states, such as Pennsylvania, North Carolina,Georgia, among others, while the third concentric circle contains only a few states, such as theoil producing states, Texas, Oklahoma, and North Dakota. The full animated representationcan be found at the author’s webpage.9

An additional advantage of the proposed framework is the possibility of recovering thestationary measures of synchronization, by using the ergodic probabilities associated with thelatent variable Vt. Chart A of Figure 2 of Appendix B plots the stationary grouping pattern,which can be interpreted as the average pattern from August 1979 to March 2013. It showsthree groups of states, corresponding to the three concentric circles: one is close to the U.S.cycle, the second is less but still close to the U.S. cycle, while the third is characterized by thestates following independent dynamics. To assess whether this result reconciles with the one inHamilton and Owyang (2012), Chart B of Figure 2 of Appendix B plots the clusters obtainedby those authors. The results show that clusters found in Hamilton and Owyang (2012) areconsistent with the grouping pattern of states found in this paper. Moreover, this result isnot only robust to the methodology employed, but also to the data used, since Hamilton andOwyang (2012) use annualized quarter-to-quarter growth rates of payroll employment, while Iuse monthly growth rates of state coincident indexes of economic activity. These facts showone of the main contributions of the proposed framework, which is to provide synchronizationmeasures that may change over time, and that can be collapsed into ergodic measures thatyield results consistent with those in previous work.

Regarding the cyclical relationship between states and the national business cycle, Ta-ble 6 reports the corresponding ergodic synchronizations, showing the range from the highestones, which are Illinois and Pennsylvania with 0.86 and 0.85, respectively, to the lowest ones,Louisiana and Oklahoma with 0.18 and 0.17, respectively. To provide a visual perspective,Chart A of Figure 3 of Appendix B plots a U.S. map with the estimates obtained in this paper,and Chart B plots the concordance pattern obtained in Owyang et al. (2005) by calculatingthe percentage of the time two economies were in the same regime, based on univariate MSmodels for each state. Although both results report high values in most of the states locatedin the east region and moderated values in a few states located in the west, the stationarysynchronization measure presents higher dispersion than the concordance, as can be seen inthe associated histograms. This comparison helps to differentiate in a more precise way thestrenghtness of cyclical relationships between the business cycles of states and the nation.

economies are in recession because they are under a regime of dependence, i.e., states 1 and 5 of S∗ab,t in Equation

(13), respectively.

4.3 Network Analysis

Recent works by Carvalho (2008), Gabaix (2011), Acemoglu et al. (2012), among others, rely onnetwork analysis to show how idiosyncratic shocks, at the firm or sectoral level, may originatemacroeconomic fluctuations, given their interlinkages. Although, such analysis primarily relieson the economy’s sectoral disaggregation, it may be interesting to assess if another type ofdisaggregation, e.g., regional, may also have significant implications on aggregate fluctuations.

The intuition behind the synchronization measure in Equation (18) relies on the fact thatif δijt is close to 1, it is likely that at time t, economies i and j are sharing the same businesscycle phases, creating a link of interdependence between them. On the other hand, if δijt is closeto 0, it means that the economies are following independent phases and thus are not linked.10

9https://sites.google.com/site/daniloleivaleon/media10Notice that the proposed synchronization modelling approach distinguishes between the state in which two

economies are in recession but their cycles are independent and just coincided, from the state where the two


Therefore, by letting H = {hi}n1 be the set of n economies taking the interpretation of nodes,hi for i = 1, . . . , n, and defining δijt as the probability that nodes hi and hj are linked at timet, the matrix Δt = 1n − Γt, can be interpreted as a weighted network of synchronization withMarkovian dynamics.11 Consequently, the cyclical interdependence of a large set of economiescan be dynamically assessed under a unified framework by relying on network analysis. It isworth noting that although the construction of Δt requires the computation of several bivariatemodels of the type in Equation (4), it may be less restrictive and involve less parameter andregime uncertainty than the computation of a framework with a similar non-linear nature butinvolving all n economies simultaneously. However, further research in this respect is needed.

To provide a glimpse of the shape that the Markov-switching synchronization network(MSYN) has taken during contractionary episodes, the charts of Figure 4 plot the correspond-ing network graph for the first month of the last four recessions. Given that the MSYN is aweighted network, in order to make the graphical representation possible, a link between nodes iand j is plotted if δijt > 0.5; otherwise, no link is plotted between them. The figure corroboratesthe grouping pattern shown in the MDS analysis, which is consistent with a relatively dispersenetwork structure during the 1981 and 1990 recessions, while showing a core and peripherystructure in the 2001 and 2007 recessions.12

The main advantage of providing a network analysis for the present framework is that all theinformation on synchronicities in the current analysis can be summarized in just one measure,the closeness centrality. There are several measures regarding the centrality of a network, butgiven that desynchronization measures are interpreted as distances, the most appropriate onefor this context is the closeness centrality.

For robustness purposes, two variations of the closeness centrality are analyzed in thissection. For each of them, it is necessary to first compute the centrality of each node,

Ct(i) =1∑

j �=i|t dt(i, j), for i = 1, 2, ..., n, (60)

where d(i, j) is the length of the shortest path between nodes i and j, which can be computed bythe Dijkstra (1959) algorithm.13 Thus, the more central a node is, the lower the total distancefrom it to all other nodes. Closeness can be regarded as a measure of how fast it will take tospread information, e.g., risk, economic shocks, etc., from node i to all other nodes sequentially.For an overview of definitions in network analysis, see Goyal (2007).

Once the dynamic centrality of each node has been computed, the information about thewhole network’s centrality can be typically assessed as follows:

CNt =

k∑i=1|t

[Ct(i∗)− Ct(i)], (61)

where i∗ is the node that attains the highest closeness centrality across all nodes at time t. Thesecond measure, consists on the average across all nodes’ centralities, Ct(i), defined by

CAt =

k∑i=1|t

Ct(i). (62)

( ), p y11The term 1n represents a squared matrix of size n with all entries equal to 1.12Notice that, although the U.S. business cycle is not included in the network analysis, only those of the

states, each chart in the figure shows a close relation with the corresponding one in Figure 3.13For example, in a set H ′ = {a, b, c} where the distances γ = 1 − δ are given by γab = 0.5, γac = 0.9 and

γbc = 0.2, the shortest path between a and c will be 0.7, since γab + γbc < γac. Thus, notice that d(a, c) doesnot necessarily have to be equal to γac.


These two measures, which provide information on the changes in the degree of aggregatesynchronization among the economies in the set H, for the present case between the states ofthe United States, can be used to investigate the relationship between regional business cycleinterdependence and aggregate fluctuations.14

One of the main findings in Hamilton and Owyang (2012) is the substantial heterogeneityacross regional recessions in the United States at the state level. How such heterogeneity couldchange over time, however, is an issue that has remained uninvestigated. The proposed frame-work is used to dynamically quantify the substantial regional heterogeneity under the unifiedsetting MSYN. The intuition behind the state’s centrality in Equation (60) is the following:if, at time t, state i is highly synchronized with respect to the rest of U.S. states, its totaldistance from them,

∑j �=i|t dt(i, j), would tend to be low and its centrality, Ct(i), to be high. If

a similar behaviour occurs with the remaining n− 1 states, the MSYN’s centrality would alsotend to take high values. This means that high global interdependence, or, equivalently, highhomogeneity of regional recessions, is associated with high values of the MSYN’s centrality CΥ

t ,for Υ = N,A.

Chart A of Figure 5 plots the network centrality, CNt , and the average centrality, CA

t .Both measures show similar dynamics, experiencing substantial changes over time that have aclose relation with the national recessions dated by the NBER, and showing some interestingfeatures. First, the centrality shows a markedly high tendency to increase some months beforenational recessions take place, implying that sudden increases in the degree of interdependenceamong states may be useful to signal upcoming national recessions. Second, once nationalrecessions have ended, the centrality measures also increase. This is because the whole economyis recovering from the recession and most of the states are synchronized, although, this time, in aexpansionary regime. Third, after this phase of recovery has ended and the U.S. economy startsits moderated expansionary path, the centrality decreases until it reaches a certain stable level,which prevails until another recession takes place and the cycle repeats. Notice that the periodswith higher heterogeneity across regional business cycles do not occur during turning points,but during periods of stable economic expansion. These three observations reveal that regionaleconomies in the United States at the state level are subject to cycles of interdependence thatare highly associated with the national business cycle, in particular, to the periods around theturning points.

The centrality measures have experienced higher levels during the 2001 and 2007 recessionsthat during the previous recessions, corroborating the core-periphery structure observed in theMDS analysis for the corresponding periods and plotted in the bottom charts of Figure 4. Thisresult discloses a change in the propagation pattern of aggregate recessionary shocks. Duringthe pre-2000 recessions, those business cycles shocks were spread mainly toward a few butrelatively large states, in terms of share of GDP, while during the post-2000 recessions, suchshocks were more uniformly and synchronously distributed across states, in particular, to theones in the core, as can be seen in the charts of Figure 3.

To address changes in the clustering pattern in a statistical rather than visual manner, Icompute the clustering coefficient of the MSYN for every time period by following Strogatz andWatts (1998), which allows the measurement of the level of cohesiveness between the businesscycle phases of U.S. states. The dynamic clustering coefficient is plotted in Figure 6, showingthat in the mid-1990s there was a significant change in the regional cohesiveness. Before thattime, the clustering coefficient followed short cycles, but after the mid-1990s, it remained almoststable at higher values, corroborating the change in the propagation of contractionary shocksthat occurred since the 2001 recession and providing evidence that the U.S. economy’s regionshave become more interdependent since the early 1990s.

14A third measure was also computed by extracting the common component among the nodes’ centralitiesusing principal component analysis. However, the results were similar to those of obtained with the averagecentrality. Therefore, they are not shown.


4.4 Explanatory Factors

In order to provide assessments about the origins of the complex interactions between the busi-ness cycles of US states, I investigate whether changes in US business cycles intra-synchronizationmay be explained by certain macroeconomic and financial factors. In particular, the interest isplace on explaining desynchronization measures, γij

t , with a set of variables that represent dis-similarities about certain features of the states, such as, sectoral composition, income, financialactivity, and fiscal policy.

If two states possess similar economic structures, both states may experience similar re-sponsiveness to business cycles shocks. Therefore, I follow the line of Imbs (2004) and use theshares in aggregate employment associated to sector l, of state i, at time t, denoted by ηil,t, tocompute a time-varying measure of industry specialization:

SEC ijt =

L∑l=1

|ηil,t − ηjl,t|. (63)

The variable SECijt measures the differences in the economic structure of states i and j over

time, and represents the one of the potential explanatory factors of changes in synchronizationto be assessed. Another potential factor is related to the wealth of states, since states withsimilar levels of household wealth may experienced similar economic fluctuations. Accordingly,I use the real median household income of state i at time t, denoted by, INC i

t , to constructthe variable:

INC ijt = | ln(INC i

t)− ln(INCjt )|, (64)

where INC ijt measures the differences in the household wealth of states. The financial structure

of states may also play an important role in explaining their business cycles synchronizationpatterns. I follow Francis et al. (2012) and use the total banking deposits of state i and timet, denoted by DEP ij

t , to measure differences in financial structures, INC ijt , analogously to

Equation (64). Finally, I consider the fiscal sector as a potential explanatory factor of businesscycles comovement and use government expenditures of state i at time t, GOV i

t , to measuredifferences in fiscal policy of states, denoted by, GOV ij

t , and computed following Equation (64).The data used in this analysis spans from 1992 until 2013, the longest available sample at

the present time, and was taken from different sources. Data on employment, at the monthlyfrequency, and data on household income, at the yearly frequency, were retrieved from theFederal Reserve Economic Data. Data on bank deposits and government expenditures, at theyearly frequency, was taken from the Federal Deposit Insurance Corporation and the CensusBureau, respectively. The objective of this section is assessing the relationship between thosefactors measuring economic differences at the state level and the measures of business cyclesdissimilarities by estimating the following panel regression:

γijt = α + β1SECij

t + β2INC ijt + β3DEP ij

t + β4GOV ijt + υij

t , (65)

for ij = 1, 2, ...,C492 , hence, the cross-sectional unit in the panel model is pairs of US states.

The estimated coefficients of Equation (65) are reported in Table 7, showing that onlydifference in sectoral composition and in household income are significant factors explainingdifferences in business cycles synchronization.15 Notice that in both cases the estimated coeffi-cient is positive, implying that increases in the similarity of the US states economic structuresare associated to increases in their business cycles synchronization. Analogously, the more sim-ilar are the states household incomes the more synchronized their business cycles tend to be.

15I use robust standard errors in all the estimations.


y yThe effect of total deposits and government expenditures on business cycles dissimilarities isalso positive, however, it is not statistically significant. It is worth mentioning that because ofpotential simultaneity bias and reverse causality, I cannot claim any causal relationship betweenfactors and dissimilarities, and only correlation statements can be claimed.

In previous sections, this paper documents an overall increase in the synchronization ofUS states since the early 1990s, as can be seen in Figure 3, implying potential instabilities inthe relationship between the synchronization and its drivers. Therefore, I investigate potentialchanges over time in those relationships, measured by the coefficients βι, for ι = 1, 2, 3, 4, ofEquation (65). In particular, I estimate Equation (65) by using only the information containedin a given year, τ , and save all the coefficients associated to each year, that is, βι,τ , for ι =1, 2, 3, 4 and τ = 1992, 1993, ..., 2013. Figure 7 plots the time-varying betas associated toeach explanatory factor, showing that the importance of sectoral composition in explainingsynchronization patterns has significantly increased during the 1990s, reaching to a stable levelthereafter. The explanatory power of household income remained positive and significant untilthe Great Recession, indicating that wealth differences across states did not play an importantrole in explaining the last simultaneous downturn of regional economies. Finally, the effectof total deposits and government expenditures on synchronization patterns is not statisticallysignificant for most of the years, which is consistent with the full sample estimates reported inTable 7. These results indicate that the main factor driving US intra-synchronization is thesimilarity of the economic structure across states. The more similar the structures, the moresimilar the responsiveness to shocks, and therefore, the higher the correlation between theirbusiness cycles.

5 Conclusions

Most of the studies on business cycle synchronization provide a general pattern of cyclicalaffiliations between economies for a given time span. However, little has been done to assesspotential pattern changes that may occur during such a time span. This paper proposedan extended Markov-switching framework to assess changes in the synchronization of cyclesby inferring the time-varying dependency relationship between the latent variables governingMarkov-switching models. The reliability of the approach to track synchronization changes isconfirmed by Monte Carlo experiments.

The proposed framework is applied to investigate potential variations in the cyclical in-terdependence between the states of the United States. There are four main findings. First,the results report the existence of interdependence cycles that are associated with NBER re-cessions. Such cycles are defined as periods characterized by low cyclical heterogeneity acrossstates, experienced during the recessionary and recovery phases, followed by longer periods ofhigh cyclical heterogeneity that occur during the phases of stable growth. Second, there aresubstantial variations in the grouping pattern of states over time that can be monitored on amonthly basis, ranging from a scheme characterized by several clusters of states to a core andperiphery structure, composed of highly and lowly synchronized states, respectively. Third,there is evidence of a change in the propagation pattern of recessionary shocks across states.Up to the 1991 recession, recessionary shocks were spread mainly toward a few large states,in terms of share of GDP. But after that, contractionary shocks were more synchronously anduniformly spread toward most of the U.S. states, implying that U.S. regions have become moreinterdependent since the early 1990s. Fourth, the main factor explaining the business cyclessynchronization patterns of US states is the similarity of their economic structure, followed byhow similar is the wealth, measured by household income, across states. The more similar thestructures and the wealth of states, the higher their business cycles synchronization.


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Table 1: Parameter values for generating processes

State 0 State 1Parameter Value Parameter Value

μ∗a,0 −1 μ∗

a,1 2μ∗b,0 −2 μ∗

b,1 4σ∗a,0 0.20 σ∗

a,1 1σ∗b,0 0.60 σ∗

b,1 3p∗a,00 0.80 p∗a,11 0.90pb,00 0.80 p∗b,11 0.90p∗00 0.80 p∗11 0.90p∗G,00 0.98 p∗G,11 0.98p∗V,00 0.98 p∗V,11 0.98

Note: The table shows the parameter values used to generate the stochastic processes yt inEquation (49) for the simulation study in Section 3.

Table 2: Performance of parameter estimations

z = 1 z = 2 z = 3 z = 4 z = 5 z = f(Vt) δ = 0.2 δ = 0.5 δ = 0.7μ∗a,0 -0.99 -0.98 -0.98 -0.98 -0.99 -0.98 -0.95 -0.97 -0.98

μ∗a,1 1.97 1.97 1.97 1.97 1.97 1.97 1.92 1.95 1.96

p∗a,11 0.89 0.89 0.89 0.89 0.89 0.89 0.81 0.80 0.79p∗a,00 0.79 0.79 0.79 0.78 0.79 0.78 0.71 0.69 0.68μ∗b,0 -1.95 -1.95 -1.95 -1.95 -1.95 -1.95 -1.91 -1.90 -1.94

μ∗b,1 3.93 3.93 3.93 3.93 3.93 3.93 3.86 3.90 3.91

p∗b,11 0.89 0.89 0.89 0.89 0.89 0.89 0.80 0.80 0.79pb,00 0.79 0.79 0.79 0.78 0.78 0.78 0.70 0.69 0.68p∗11 0.89 0.88 0.89 0.88 0.89 0.88 0.80 0.79 0.78p∗00 0.77 0.76 0.76 0.76 0.76 0.76 0.69 0.67 0.66σ∗a,0 0.26 0.26 0.26 0.26 0.26 0.26 0.34 0.30 0.27

σ∗a,1 0.75 0.76 0.75 0.77 0.76 0.77 0.94 0.88 0.80

σ∗b,0 1.01 1.01 1.01 1.01 1.02 1.02 1.08 1.04 1.02

σ∗b,1 3.03 3.05 3.07 3.05 3.06 3.04 3.26 3.16 3.16

p∗G,11 0.96 0.92 0.95 0.93 0.95 0.93 0.95 0.95 0.96p∗G,00 0.94 0.96 0.95 0.96 0.94 0.94 0.96 0.96 0.96p∗V,11 - - - - - 0.96 - - -p∗V,00 - - - - - 0.96 - - -

Note: The entries in the table report the average of the estimated parameter values throughthe 1,000 replications for different numbers of synchronization changes, z.


Table 3: Performance of regime inferences

z = 1 z = 2 z = 3 z = 4 z = 5 z = f(Vt) δ = 0.2 δ = 0.5 δ = 0.7QPS(Sa,t) 0.02 0.02 0.02 0.02 0.02 0.02 0.13 0.12 0.13QPS(Sb,t) 0.01 0.02 0.01 0.02 0.01 0.02 0.13 0.12 0.13QPS(Gt) 0.03 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.03QPS(Vt) 0.02 0.04 0.04 0.05 0.06 0.08 0.06 0.08 0.07

δ̄ - - - - - - 0.27 0.55 0.75

Note: The entries in the table report the average of the Quadratic Probability Score associatedwith the state variables through the 1,000 replications for different numbers of synchronizationchanges, z, and levels of imperfect synchronization δ. The term δ̄ makes reference to the averageestimated synchronization over a regime of imperfect synchronization.

Table 4: Dynamic synchronization estimates between New York and Texas

Mean Median Std. Dev.μny,0 -0.10 -0.10 0.04μny,1 0.38 0.38 0.03σ2ny,0 0.02 0.02 0.02

σ2ny,1 0.09 0.08 0.02

pny,11 0.98 0.98 0.00pny,00 0.94 0.94 0.02μtx,0 -0.05 -0.05 0.02μtx,1 0.43 0.43 0.02σ2tx,0 0.01 0.01 0.00

σ2tx,1 0.08 0.08 0.01

ptx,11 0.98 0.98 0.00ptx,00 0.93 0.94 0.02σny,tx,0 0.00 0.00 0.00σny,tx,1 0.03 0.03 0.02p11 0.98 0.98 0.00p00 0.93 0.93 0.02pG,11 0.95 0.97 0.03pG,00 0.95 0.97 0.04pV,11 0.94 0.94 0.02pV,00 0.97 0.98 0.00

Note: The selected example presents the case of two states with high and similar shares of U.S.GDP, New York with 7.68% and Texas with 7.95%.


Table 5: Dynamic synchronization estimates between California and Vermont

Mean Median Std. Dev.μny,0 0.03 0.03 0.02μny,1 0.34 0.34 0.01σ2ny,0 0.01 0.01 0.00

σ2ny,1 0.23 0.21 0.07

pny,11 0.97 0.98 0.00pny,00 0.95 0.95 0.01μtx,0 0.00 0.00 0.03μtx,1 0.37 0.37 0.03σ2tx,0 0.05 0.05 0.00

σ2tx,1 0.48 0.45 0.16

ptx,11 0.97 0.97 0.00ptx,00 0.94 0.95 0.01σny,tx,0 0.00 0.00 0.00σny,tx,1 0.18 0.17 0.09p11 0.97 0.98 0.00p00 0.95 0.95 0.01pG,11 0.93 0.94 0.04pG,00 0.96 0.97 0.03pV,11 0.99 0.99 0.00pV,00 0.87 0.88 0.05

Note: The selected example presents the case of the states with the highest and the lowestshares of U.S. GDP, California with 13.34% and Vermont with 0.18%.


Table 6: Stationary synchronization between individual states and the entire United States

State Sync State Sync State SyncAlabama 0.72 Maine 0.71 Ohio 0.80Arizona 0.59 Maryland 0.76 Oklahoma 0.17Arkansas 0.79 Massachusetts 0.69 Oregon 0.78California 0.74 Michigan 0.69 Pennsylvania 0.85Colorado 0.75 Minnesota 0.81 Rhode Island 0.59Connecticut 0.70 Mississippi 0.68 S. Carolina 0.80Delaware 0.61 Missouri 0.73 S. Dakota 0.46Florida 0.76 Montana 0.21 Tennessee 0.73Georgia 0.74 Nebraska 0.50 Texas 0.42Idaho 0.65 Nevada 0.52 Utah 0.64Illinois 0.86 N. Hampshire 0.59 Vermont 0.67Indiana 0.81 New Jersey 0.74 Virginia 0.81Iowa 0.54 New Mexico 0.52 Washington 0.77Kansas 0.72 New York 0.80 Wisconsin 0.74Kentucky 0.75 N. Carolina 0.81 W. Virginia 0.69Louisiana 0.18 N. Dakota 0.21 Wyoming 0.27

Note: The table reports the stationary synchronization for the period August 1979 to March2013. These estimates correspond to the ergodic probability that the phases of the state businesscycles and U.S. business cycles are the same, i.e., Pr(Vt = 1). The index used to measure thenational business cycle is the Chicago Fed National Activity Index (CFNAI).

Table 7: Panel Regression

(1− δijt )Sectorial Composition 1.223∗∗∗

(18.97)Real Median Household Income 0.151∗∗∗

(3.94)Total Deposits 0.00761

(1.68)Government Expenditure 0.0136

(1.82)Constant 0.363∗∗∗

(20.68)Observations 281640

Note: The table reports the estimates of βι in Equation (65). t statistics are reported inparentheses. Asterisks are defined as, ∗ for p < 0.05, ∗∗ for p < 0.01, ∗∗∗ for p < 0.001.


Figure 1: Simulation of changes in synchronization of cycles

(a) z=1

-4

-2

0

2

4

6

0

1

50 100 150 200 250 300 350 400

Y_a Y_b V

0.0

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50 100 150 200 250 300 350 400

P(V=1) V

-6

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Y_a Y_b G

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P(D=1) D

(b) z=2

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P(G=1) G

Note: The figure plots one simulation for the cases of 1, 2 and 3 changes in the synchronicityof cycles. For each case, the top panels plot the generated pair of time series along with theindicator variable of synchronization changes. The two middle panels plot the probabilitiesof a low mean regime associated with each time series, along with the indicator variable asreference. The bottom panels plot the estimated dynamics of the indicator variable along withthe real one.


Figure 2: Dynamic synchronization between selected states

(a) New York and Texas

0.0

0.2

0.4

0.6

0.8

1.0

80 85 90 95 00 05 10 15

Recession Probability of NY

0.0

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80 85 90 95 00 05 10 15

Recession Probability of TX

0.0

0.2

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80 85 90 95 00 05 10 15

Dynamic Synchronization of NY and TX

0.0

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80 85 90 95 00 05 10 15

High Volatility Probability of NY and TX

(b) California and Vermont

0.0

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80 85 90 95 00 05 10 15

Recession Probability of CA

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80 85 90 95 00 05 10 15

Recession Probability of VT

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Dynamic Synchronization of CA and VT

0.0

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High Volatility Probability of CA and VT

Note: The figure plots the output estimation for two selected pairwise models. Chart A plotsthe probability of recession for New York and Texas along with their dynamic synchronization.Chart B plots the probability of recession for California and Vermont along with their dynamicsynchronization. Shaded areas correspond to NBER recessions.


Figure 3: Dynamic synchronization maps of U.S. states across recessions

(a) (b)

(c) (d)

Note: Each chart in the figure plots the dynamic multi-dimensional scaling map based onthe synchronization distance of the business cycle of U.S. states for different periods. Thedistances are normalized with respect to the U.S. national economic activity, the grey pointin the centre. The size of the points refer to the GDP share of the corresponding state.If two states are placed in the same concentric circle, they are equally in sync with theUnited States. The full animated version of the synchronization mapping is available athttps://sites.google.com/site/daniloleivaleon/media.


Figure 4: Synchronization network of the U.S. states across recessions

(a) (b)

(c) (d)

Note: The figure plots the interconnectedness in terms of synchronization between the businesscycle phases of U.S. states. Each node represents a state and each line represents the linkbetween two states, which takes place only if Pr(V t = 1) > 0.5. The full animated version canbe found at https://sites.google.com/site/daniloleivaleon/media.


Figure 5: Dynamic closeness centrality of the U.S. synchronization network

(a) Closeness centrality

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1980 1985 1990 1995 2000 2005 2010 2015

(b) Average closeness centrality

3.6

4.0

4.4

4.8

5.2

5.6

6.0

6.4

6.8

7.2

1980 1985 1990 1995 2000 2005 2010 2015

Note: Chart A and Chart B plot the closeness and average closeness centrality measures ofthe Markov-switching synchronization network, respectively. The solid line plots the networkcloseness centrality defined in Equation (61) and the dotted line plots the average centrality, asdefined in Equation (62). Left axis of Chart B are in percentages units, and left axis of ChartA are in regular units. Shaded bars refer to the NBER recessions.

Figure 6: Dynamic clustering coefficient of the U.S. synchronization network

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

1980 1985 1990 1995 2000 2005 2010 2015

Note: The figure plots the time-varying clustering coefficient of the Markov-Switching Synchro-nization Network for U.S. states. Shaded bars refer to the NBER recessions.


Figure 7: Time-varying relationship between explanatory factors and synchronization

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

92 94 96 98 00 02 04 06 08 10 12

95% CI Sectoral Composition

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

92 94 96 98 00 02 04 06 08 10 12

95% IC Real Median Household Income

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

92 94 96 98 00 02 04 06 08 10 12

95% CI Total Deposits

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

92 94 96 98 00 02 04 06 08 10 12

95% CI Government Expenditure

Note: The figure plots the estimated coefficients that measure the time-varying relationshipbetween the synchronization and its potential explanatory factors. Dashed lines represents the95% confidence interval.

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